However, if the image has a caption, it shows up underneath its image, throwing off my neat little grid of images. I don't want to turn captions completely off or not use them at all, because after the grid and below the main 960x540 image, I want to do an "Image Summary" kind of section, with the filename, original size, caption, description, etc.

Is there a way to turn off captions in the above code? The gallery shortcode section of the WordPress Codex doesn't have anything on it.